By Gko. E. Woodward, Civll. And Landscape Engineer, St Park Row, New York.

The process of laying out circular and elliptical flower-beds is familiar to many, but the rules hitherto given for laying out irregular figures having curved boundary lines have been exceedingly troublesome, and, unless one is possessed of some ingenuity, are of but little value. The usual practice is to set out the curves by ordinates or perpendiculars from a straight line, these ordinates being either measured on a plan by a scale, or the length of each calculated, and involving more of a practical knowledge of mathematics than the method we propose to explain. It is quite necessary that all operations of this kind be made as simple as possible, and that the most beautiful and graceful forms be adopted, as they involve no more labor in laying them out than those of the most common-place character. We can well imagine the ten-fold interest that gathers around the pursuit of & higher range of art which, at even less expenditure of time and money, develops for more beautiful and attractive forms and combinations.

As all forms of the beautiful are bounded by strictly mathematical curves, it is proper that we should make use of the resources of mathematics in reproducing beautiful lines and forms, as they give the power to execute rapidly and with a certain result. In designing flower-beds we must exercise our taste in the combination of curve lines, so that the entire outline or form of the figure be agreeable; and although we are dealing with the most beautiful of all lines considered separately, it does not follow that a combination of them must necessarily be beautiful; but very beautiful forms can be made by any one who tastefully arranges them.

In the diagram we show all the auxiliary lines, the better to explain the prin ciple. It will be observed that the figure is a combination of circular curves struck from different centers, and that the union of each is perfectly graceful, not the slightest abruptness or departure from a harmoniously flowing line. This is produced by the well-known system of drawing the curves so that they are tangent to, or touch each other; and in no other manner can two circular curves unite so that the point of meeting shall be absolutely graceful; the centers from which each curve is described and the point of contact must be in the same straight line.

As a matter of economy as well as gratification, it is better to study out the design, and the manner in which it shall be executed, on paper. Thus, having decided on the figure shown in the diagram, from the point A, with a divider and to a scale, describe the are G H. On the line from A to H must be the center of the next curve, which we fix at B; from B describe the arc HI. On the line B I must be the center of the next curve, which we fix at C, and from C as a center describe the arc I J. Now, if we examine the figure, it will be seen that the circle of which B is the center lies wholly within that of which A is the center, and touches it at one point only, at H; the circle of which C is the center lies wholly within that of which B is the center, and touches it only at the point I; and that the centers of each two curves that unite and their point of contact are in the some straight line, as A, B, H, and B, C, I. At the point J we reverse the curve, and to do so draw a line from C through J, and produce to D, the center of the next curve, and which must be in this line; from D, with a radius D J, describe the circle of which D is the center; at the point J this circle then touches or is tangent to the circle of which C is the center, and C, J, and D are in the same straight line.

The curves K L and L G are struck from independent centers, and give variety to the figure. If desirable to inclose the whole figure with curved lines, connect the last curve made from the center D with the first one made from the center A by drawing a line from A through D, and producing it to the circumference of the circle of which A is the center; then bisect that part of the line between the circumference of which D is the center and that of which A is the center, and describe the circle of which £ is the centre, which is tangent to both circles A and D. This would form a second figure, and leave a third cut off. A fourth and fifth figure can be made by cutting the original figure in two with part of the circumference of the circle of which B is the center, as shown by the heavy dotted lines; and a sixth figure would be left by removing from the original figure the circle of which C is the center. Thus, by this process, we can make six or more different figures, either one of which has a good form, and we may say that the design of most of them was purely accidental. This mode may be carried almost to infinity, and one could hardly fail to find on every trial a form which might be claimed as original.

A half hour's trial with a ruler and dividers is sufficient to make any one an expert.

Having decided on the original design, all that is necessary is to transfer the centers A, B, C, D, and the initial point G, to the ground, and in the same relative positions as on the plan. They might be set out on a base line; thus, draw on the plan a straight line running between the five points named, and measure with a scale the perpendicular distance to each point, and from the same line on the ground set them out; then from A, with a radius A G, (which should be a metallic tape or chain to prevent stretching,) describe the curve G H; II being in line with A B, is easily found; then from B describe I H; I will be in line with C B, and wo on. The centers should be preserved, so that at any time when grass grown, the verge can be cut clean and accurate in the same manner as practiced in beds of single circles; an advantage that will be appreciated by those who have an artistic eye.

This process of joining curves of greater or less radius, or reversing them, is applicable to roads and walks, and is the only, mode by which curves flow gracefully into each other, and any plan which does not embrace this principle is defective in artistic excellence. We do not mean to say that curves of long radii are struck from centers, but that curves of different radii, when so used, should be tangent to each other, whether laid out from centers or on the circumference. We do say that compound circular curves are practically identical with any curve that can possibly be made use of in any department of landscape adornment, and that there is no curve known, or gracefully flowing line, but what is rigidly mathematical. Will some one give us an example to the contrary 1

[We can not help adding a word of commendation here. We regard this as one of the very best and most useful articles that Mr. Woodward has written on this subject; he has here made mathematics available to every body. The principle illustrated is simple, and we know it to be beautiful in its results, while it is capable of infinite diversity. The reader will do well to study it thoroughly. - Ed].